Weighted geometric mean of n-operators with n-parameters

We shall consider a weighted geometric mean of n-operators with n-parameters. It is based on the geometric mean defined in our previous paper [C. Jung, H. Lee, T. Yamazaki, On a new construction of geometric mean of n-operators, Linear Algebra Appl. 431 (2009) 1477–1488]. Then we shall show that its weights can be obtained by simple forms in the commutative operators case. Some properties of the weighted geometric mean are obtained.